Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C4×C5×#472 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(6,10)(7,14)(8,11)(13,18)(15,16)(17,19);(5,6,11,18,9,13,8,10)(7,12,14,19,16,20,15,17);(5,7)(6,12)(8,15)(9,16)(10,17)(11,14)(13,20)(18,19)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14541);